(Jim Schultz/The Record Searchlight via AP). Crime tape blocks off Rancho Tehama Road leading into the Rancho Tehama subdivision south of Red Bluff, Calif., following a fatal shooting on Tuesday, Nov. 14, 2017.

(CNN/Meredith) -- The gunman in a deadly shooting rampage in Northern California was identified as Kevin Neal, Tehama County Assistant Sheriff Phil Johnston said Wednesday. Police shot him dead Tuesday after he killed four people in a string of shootings. His wife's body was found under the floor at his home, bringing the death toll to five.

The fatal shooting of his wife may have led to Neal's deadly outburst, Johnston said.

Before police shot him dead, the man killed four people -- including the neighbor -- and injured at least 10 others in a string of shootings that spanned at least seven sites in the small community of Rancho Tehama, west of Corning, police said. Those locations include a locked elementary school that the shooter fired into from the outside, injuring a boy. The police haven't released the gunman's name.

The shootings started in his neighborhood shortly before 8 a.m. Tuesday, where he killed a woman neighbor who had a restraining order against him, Tehama County Assistant Sheriff Phil Johnston said. The man had been arrested and charged in January on suspicion of assaulting her with a deadly weapon, Johnston said.

He then stole a pickup truck and went on a rampage in town, shooting at homes, motorists and eventually the school, apparently at random, Johnson said.

"This guy was bent on ... driving by residences and arbitrarily shooting at them," the assistant sheriff said.

Still, Johnston said it "could have been so much worse." Wearing a tactical vest and armed with extra magazines, the gunman attempted to enter classrooms at Rancho Tehama Elementary School.

Ramming through the school's gate

The gunman arrived at the small elementary school -- about 2 miles from his home -- just before classes were to begin. The staff, having heard gunfire nearby or been alerted to it, had already begun to take action.

Doors were locked and students rushed inside, where they hit the floor underneath desks and tables.

The shooter rammed the stolen pickup through the school's locked gate, walked into the schoolyard and fired his rifle through windows and walls.

He could not get inside and left six minutes later, apparently frustrated, Johnston said.

One student was in stable condition after being wounded by gunfire, the school district said. Others were hurt by flying glass.

Before reaching the school, the gunman had fired from his vehicle at passing motorists and homes, Johnston said. At one point he shot and severely injured a woman who was driving her three children to school.

The woman was being treated for life-threatening injuries. A boy who was in the vehicle was hit by gunfire or shrapnel but was not seriously wounded, according to Johnston.

"She has no clue who he was," the official said.

After leaving the school, the shooter intentionally crashed the pickup into a car. He exited the truck and gunned down at least one person there, Johnston said.

He then stole someone else's sedan and "went back on his rampage," Johnston said.

Eventually, two police officers encountered the gunman at an intersection. They returned fire, killing him, the assistant sheriff said.

Aside from the first shooting in his neighborhood, the killer appeared to be shooting people at random, Johnston said, and did not seem to have any connections to the school either.

"This is an individual who armed himself, I think with the motive of getting even with his neighbors, and when it went that far (that someone was killed) he just went on a rampage," Johnston said.

"I have to say this incident, as tragic and as bad as it is, could have been so much worse," he said, applauding the quick thinking of the school staff.

Mom raced back to school to warn people

A fearless mother may have also helped warn staff at the school about the danger.

Sara Gonzalez said she had dropped her daughter off at the school when the gunman attempted to shoot her as their vehicles passed. She said thinks the gunman's windshield stopped the bullet.

She turned her vehicle around.

"I went to my daughter's school and started honking, making people know what was going on," she said.

Gonzalez said the gunman fired at her as she tried to get out and find her daughter. She saw the man's face but he said nothing.

He said he ran over to the school when he heard the crack of a semi-automatic rifle.

"When I get there, the dude was shooting through the windows," he told the Sacramento TV station. "And I said, 'Hey why don't you shoot this way instead.'"

Sanders said a bullet grazed his right forearm.

Inside the school, the children tried to stay low.

Gonzalez's daughter, Arianna Ibarra, was on the floor with her fourth-grade class.

"Our teacher told us to go under our desk and keep flat in case he comes inside," Arianna said.

The teacher blocked the door with a computer, the 10-year-old said.

Three firearms among the weapons recovered

It was unclear whether the gunman went to the school looking for someone he knew.

"We have not connected him with any one person in the school," Johnston said.

A semi-automatic rifle and two handguns believed to be used by the gunman have been recovered, according to Johnston. There is another gun that police have yet to recover in one of the stolen vehicles.

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